Vague Language and Context Dependence∗

Wooyoung Lim† and Qinggong Wu‡

April 19, 2018

Abstract

In this paper, we broaden the existing notion of vagueness to account for linguisticambiguity that results from context-dependent use of language. This broadened notion,termed literal vagueness, necessarily arises in optimal equilibria in many standardconversational situations. In controlled laboratory experiments, we find that, despitethe strategic complexity in its usage, people can indeed use literally vague language toeffectively transmit information. Based on the theory and the experimental evidence,we conclude that context dependence is an appealing explanation for why language isvague.

Keywords: Communication Games, Context Dependence, Vagueness, Laboratory Ex-perimentsJEL classification numbers: C91, D03, D83

∗We are grateful to Andreas Blume, Itzhak Gilboa, Ke Liu, Joel Sobel and Alistair Wilson for their valu-able comments and suggestions. For helpful comments and discussions, we thank seminar and conferenceparticipants at the SURE International Workshop on Experimental and Behavioral Economics (Seoul Na-tional University). This study is supported by a grant from the Research Grants Council of Hong Kong(Grant No. GRF-16502015).

†Department of Economics, The Hong Kong University of Science and Technology. Email:wooyoung@ust.hk

‡Department of Decision Sciences and Managerial Economics, Chinese University of Hong Kong. Email:wqg@baf.cuhk.edu.hk

1 Introduction

Language is vague, as this very sentence exemplifies (what does “vague” mean here, afterall?), but we do not quite know why. On one hand, vagueness decreases accuracy andcauses confusion; on the other hand, there seems to be no obvious benefit to vaguenessthat justifies its prevalence. Indeed, as Lipman (2009) forcefully argues, vagueness hasno efficiency advantage in a game-theoretic model of communication if agents are rational.Thus, Lipman (2009) suggests, it seems that the presence of vagueness can be explainedonly if some of the agents are not entirely rational.

In this paper, we also attempt to explain vagueness. Rather than following the agendaof Lipman (2009), we instead maintain the rationality assumption but broaden the notionof vagueness. Language is defined as vague1 if words are not divided by sharp boundaries—the set of objects described by a word may overlap with that described by another word.For instance, “red” and “orange” overlap in describing reddish-orange colours. We findthis definition restrictive in that it rules out languages that would appear to people asvague. In particular, we observe that a word’s operational meaning often depends on thecontext in which the word is used. Even if words are divided by sharp boundaries in everycontext, these boundaries may vary with the context and become fuzzy if the context isnot clear to the listener, thereby creating a sense of vagueness, despite the fact that thelanguage is technically not vague. In short, context-dependent use of words can cause alanguage to be perceived as vague. Based on this observation, we define a new notion, lit-eral vagueness, which accommodates not only conventional vagueness but also contextdependence.

We show that literal vagueness can, and often does, have an efficiency advantage, inthe sense that the optimal language in the standard model of information transmissionis often literally vague. This is particularly the case when the information to be trans-mitted has some “unspeakable” aspects that are not explicitly describable by the availablevocabulary. These aspects constitute the context. A literally vague language is more effi-cient because it is more capable of conveying information about the context than a literallyprecise language is.

Thus, a literally vague language is more efficient—but is this why everyday languageis (literally) vague? We should note that a literally vague language might be difficult touse in reality because it requires complex strategic thinking. Our proposition, that theefficiency advantage of literal vagueness provides some explanation for why language isvague, would be greatly weakened if the strategic hurdle were so high that people could

1The formal game-theoretic version of this definition is first given in Lipman (2009).

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not effectively communicate with a literally vague language. With this concern in mind,we conducted a series of experiments to explore whether efficient literal vagueness would,as theory predicts, emerge in real conversations. We found that indeed it did.

In the experiments, we considered a conversational environment in which two speakersspeak sequentially to a listener and the way that the later speaker uses his words mayrely on what the earlier speaker said. In this simple environment in which the optimallanguage is literally vague, we observed that subjects indeed tended to communicate insuch a way. Several variations of the environment with varying degrees of complexity incoordinating on the optimal, literally vague language provide further supporting evidence.

In this paper, we suggest an explanation for why language is vague, and report evidencesupporting the theory. However, we do not claim to have completely answered the question,and we are aware of the limits of the explanatory power of literal vagueness. On thetheory side, we observe that language can be vague even when context dependence is nota concern, especially when full rationality is not guaranteed. On the experiment side, theobservation that people can communicate with a literally vague language does not identifythe efficiency advantage of literal vagueness as the explanation for the omnipresence ofvague languages. Nevertheless, we believe that our exercise is a worthwhile step towardsunderstanding vagueness.

In Section 2, we discuss the theoretical aspects of vagueness and literal vagueness.Section 3 describes the experimental design. In Section 4, we report the experimentalfindings. The related literature is discussed in Section 5. Section 6 concludes. In theAppendix, we include additional details on the experiments as well as some figures andtables.

2 Vagueness and Literal Vagueness

In this section, we first discuss the formal game-theoretic definition of vagueness. Then,we illustrate why this definition is restrictive and introduce a broader notion: literalvagueness. Finally, we show that the optimal language in many communication situa-tions must be literally vague.

Vagueness

In the standard communication model, a speaker who is informed about the state of theworld ω ∈ Ω wishes to transmit this information to an uninformed listener. The speaker

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transmits the information by sending the listener a message from a message setM . Giventhe message, the listener chooses an alternative from a set A. For simplicity, assume thatΩ, A and M are finite. Conditional on state ω and the listener’s choice a, both playersreceive the same payoff of u(ω, a).2

In this setting, the language is the speaker’s message-sending strategy λ ∶ Ω→∆(M),where ∆(M) is the set of all lotteries over M . As the speaker observes that the state is ω,he uses the lottery λ(ω) to draw a message to send to the listener.

The game-theoretic notion of what it means for a language to be vague or precise isdue to Lipman (2009): Language λ is vague if it is a non-degenerate mixed strategy andprecise otherwise. This definition captures the essence of vagueness vis-a-vis precision: Ifλ is a non-degenerate mixed strategy, then multiple messages may be used to describe thesame state. Thus, the boundaries between messages are not always sharp. In contrast, ifλ is a pure strategy, then no such overlapping exists and the boundaries between messagesare sharp.

Given language λ, let πλ(m) denote the set of states in which message m is sent withpositive probability. It is clear that Πλ ∶= πλ(m)m∈M is a covering of Ω, and moreover, Πλ

is a partition of Ω if λ is precise. In our further analysis, this partition Πλ will serve as auseful representation of a precise language λ.

The central thesis of Lipman (2009) is that vagueness has no efficiency advantage overprecision because there is always a precise language among the optimal equilibrium lan-guages in any communication setting.

Literal Vagueness

The following example demonstrates that the definition of vagueness provided above doesnot capture a certain degree of vagueness in speech.

Example 1. In a conference, a presenter, Bob, asks an audience member, Alice, “How wasmy talk?” Alice’s reply depends on 1) whether she likes Bob’s research (L) or not (NL) and 2)whether she has time for further conversation (T ) or is in a hurry to the next session (NT ).If Alice does not have time, she says “It was interesting.” (l). If she has time, she says “Itwas interesting” if she likes the talk or “Here are a few things you could try” (nl) if she isless certain about Bob’s research.

Is Alice’s language vague? This example is formalized with the state set Ω = L,NL ×

2This model, particularly the common interest assumption, follows Lipman (2009). When there is aconflict of interest, the role of vagueness as an efficiency-enhancing device is well understood in the literature(see, e.g., Blume, Board, and Kawamura (2007) and Blume and Board (2014b)).

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T,NT and the message set M = l, nl. Alice’s language is precise by definition becauseshe uses the message nl with certainty in state (NL,T ) and message l with certainty inother states. Her language nonetheless “sounds” vague: If she says “It was interesting”,either she may like Bob’s research or she may not and is just in a hurry to end the conver-sation.

If Alice’s language is precise, where does the sense of vagueness originate? Observethat the messages “It was interesting” and “Here are a few things you could try” literallyconcern Bob’s research only and are irrelevant to Alice’s hurriedness. Alice’s language isprecise with respect to the states, but is not precise with respect to how she likes Bob’sresearch. The sense of vagueness comes from the language’s failure to unambiguouslydescribe the particular aspect of the state the messages are literally intended to describe.

Languages such as Alice’s are deemed to be precise by the formal definition becausein the standard model of conversation, Ω and M are abstract sets that lack structure andrelation. Within the model, there is no linguistic convention, and hence, no exogenousliteral interpretation exists apart from the speaker’s idiosyncratic use of the messages.In other words, interpretation is purely endogenous. In contrast, a word in a naturallanguage is literally intended to describe a particular aspect of the world, e.g., colour orheight. A sense of vagueness could occur if there is inconsistency between the literalinterpretation and an individual’s idiosyncratic language use, as is the case in Example1.

This observation motivates us to modify the standard setting to incorporate an ex-ogenous literal interpretation. Suppose the state set now has a two-dimensional productstructure such that Ω = F ×C. For any ω = (f, c) ∈ Ω, f is called the feature of the state andc the context. Messages in M literally describe features but not contexts. In Example 1,we have F = L,NL, C = T,NT and M = l, nl because the messages literally concernonly Alice’s attitude towards Bob’s research.

A new notion of vagueness can thus be defined as follows. A language λ is said to beliterally vague if it satisfies one of the following two conditions:

L1. λ is vague.

L2. There exist distinct f, f ′ ∈ F and distinct c, c′ ∈ C such that λ(f, c) = λ(f ′, c′) andλ(f, c) ≠ λ(f, c′).

Otherwise, we say λ is literally precise.33The notions of context and literal vagueness in our framework resemble the “prior collateral information”

in Quine (1960) and the “indeterminacy of indicative meaning” of Blume and Board (2013), respectively.

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According to L1, a vague language is literally vague. L2 is novel: It implies that bound-aries between messages with respect to features vary with the context. To see this, supposethat λ is precise but satisfies L2. The same message m describes both states (f, c) and(f ′, c′), while a different message m′ describes state (f, c′). Observe that m and m′ overlapin describing the feature f , although these two messages do not overlap in describing thestates.

Returning to Example 1, Alice’s language is literally vague because λ(L,T ) = λ(NL,NT )

= l yet λ(L,T ) ≠ λ(L,NT ) = nl. The boundary between messages l and nl is not sharp re-garding Alice’s attitude towards Bob’s research, as when Alice has doubts (NL obtains),she sometimes uses nl vis-a-vis l to clearly express them but sometimes uses nl regardlessof her attitude.

The following lemma presents a useful representation of a literally precise languageusing the partitional knowledge structure it induces for the listener.

Lemma 1. A precise language λ is literally precise if and only if every π ∈ Πλ is the Cartesianproduct of a subset of F and a subset of C.

Proof. Take a precise language λ. The “if” direction is immediate. Suppose λ is literallyprecise. For any π ∈ Πλ, let F denote the projection of π to F and C the projection of π toC. Pick any f ∈ F and c′ ∈ C. For any c ∈ C where (f, c) ∈ π and f ′ ∈ F where (f ′, c′) ∈ π, wehave λ(f, c) = λ(f ′, c′), which in turn implies λ(f, c′) = λ(f, c) because L2 is not satisfied. Itfollows that π is the product of its projections to F andC, proving the “only if” direction.

This representation has a clear interpretation: In a literally precise language, the setof features described by a message remains the same regardless of the context.

Literal Vagueness and Context Dependence

We see that literal vagueness prevails only if the set of features described by a messageis context dependent. However, context dependence does not necessarily imply literalvagueness. For example, the same person may be referred to as Mark Twain or SamuelLanghorne Clemens depending on the context. This type of context dependence not nec-essarily cause literal vagueness, because distinct messages may simply convey variationin the context but not variation in the feature. In situations of context dependence thatExample 2 and its subsequent discussion further elaborate how one can relate our discussion of literalvagueness to that provided by Blume and Board (2013). For additional discussion, see the literature reviewin Section 5.

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cause literal vagueness, the speaker chooses the same message to describe different setsof features depending on the context.

Context dependence and, hence, literal vagueness arise from the need to convey in-formation not only about features but also about contexts when the available languageis limited, in the sense that it literally describes features only.4 A natural question im-mediately follows: Why not use a richer language that also includes words that literallydescribe the context? Below, we provide a few reasons.

The first reason is that the environment of the conversation restricts the language toonly describe features, particularly if the sender responds to a question framed in a certainway. Example 1 illustrates this case: Alice responds to Bob’s question about his researchand, therefore, appropriately uses messages that literally concern only this aspect.5

Second, in certain situations, a richer language does not exist. For example, the com-mon language shared by people from different linguistic backgrounds is limited. Peoplewith different native tongues can use a facial expression in person or emojis , / emoti-cons :-) on the internet as a common language to describe an emotion, but this commonlanguage lacks terms that describe any other aspect of the world.

The third reason is that the context is sometimes meta-conversational,6 i.e., it includesinformation about the environmental parameters of the conversation itself. This informa-tion is usually not explicitly discussed in a conversation. Here, we give two elaboratingexamples.

Example 2. Alice wishes to describe to her tailor, Bob, the colour she has in mind for hernext dress. Alice may have a small vocabulary for colours that has only coarse terms suchas “red” or “blue”, or she may have an extensive vocabulary that includes words to denotesubtle colours such as “burgundy” and “turquoise”. Alice’s vocabulary is unknown to Bob.

If Alice’s vocabulary is rich, she uses “burgundy” for burgundy-red and “red” for red-red;if her vocabulary is poor, she uses “red” for both burgundy-red and red-red.

In this example, the subject matter of the conversation is colour (the feature) and the4It is not uncommon in the literature to consider a limited number of messages available for commu-

nication. For example, Sobel (2015) takes a common-interest version of Crawford and Sobel (1982) andinvestigates how the number of available messages shapes the optimal language. Similarly, Cremer, Gari-cano, and Prat (2007) study optimal codes within firms when the set of words is finite. For a more recentdiscussion, see Dilme (2017), who covers the limit case of both Sobel (2015) and Cremer, Garicano, and Prat(2007).

5This notion is related to the indirect speech act in linguistics. Speaking directly about context whenanswering a question only about features may be awkward and is thus more costly. The degree of awkward-ness an individual feels may differ, and those who feel more awkward may use only vague language. Formore details, see Pinker (2007).

6The prefix “meta-” means “on a higher level”.

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environmental parameter of the conversation is Alice’s vocabulary (the context). Becausethey are colour terms, the messages are literally intended to describe the colour, not Alice’svocabulary. It is easy to verify that Alice’s language is literally vague.

This example is based on the model studied in Blume and Board (2013). In their model,the available messages may vary depending on the speaker’s language competence. Ob-viously, the essence of their model can be embedded into ours by interpreting languagecompetence as the context, and it follows that their notion of “indeterminacy of indicativemeaning” translates into literal vagueness.

Example 3. Alice wishes to describe Charlie’s height to Bob so that Bob can recognizeCharlie at the airport and pick him up. Moreover,

• Alice knows whether Charlie is an NBA player.

• Bob may or may not know whether Charlie is an NBA player.

• Alice may or may not know whether Bob knows whether Charlie is an NBA player.

Alice describes whether Charlie is “tall” or “short”. If she believes Bob knows that Charlie isan NBA player, then she says Charlie is “tall” only if his height is above 6 foot 10, whereasif she believes Bob does not know that Charlie is an NBA player, then she says Charlie is“tall” only if his height is above 6 foot 2.

In this example, “tall” and “short” are messages that literally concern Charlie’s height(feature), where Alice’s belief about whether Bob knows that Charlie is an NBA player isthe environmental parameter of the conversation and, hence, the context. Clearly, Alice’slanguage is literally vague.

A more formal model for the example is as follows: The listener’s knowledge about thepayoff-relevant state set is not common knowledge. In particular, before the conversation,the listener may receive with some probability an informative private signal that narrowsdown the possible set of payoff-relevant states. Moreover, the speaker may also receivea private (possibly noisy) signal that tells him whether the listener has been given thatinformative signal. It is straightforward to construct a typical decision problem underwhich, in the optimal equilibrium, the meaning of a message from the speaker dependson the signal that he receives. If we do not incorporate the speaker’s signal as part of thestate, then the optimal language is vague. However, in the enriched model in which thespeaker’s signal is considered the context and the payoff-relevant state is considered thefeature, the optimal language is not vague but is literally vague.

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Example 3 and the general model described in the previous paragraph resemble themodel studied by Lambie-Hanson and Parameswaran (2015) in which the speaker and thelistener receive correlated signals about the listener’s ability to interpret messages.

Finally, we show an example demonstrating that context may arise endogenously in asequential communication procedure that resembles information cascade. When multiplespeakers speak sequentially, the way later speakers talk may rely on what earlier speak-ers have said. Earlier messages become the context on which later messages depend –the context of the bilateral conversation between a later speaker and the listener is thenendogenously generated in the larger-scale multilateral conversation.

Example 4. Alice and Bob interview a job candidate. Alice observes the candidate’s ability,which is evaluated by a number A. Bob observes the candidate’s personality, which isevaluated by a numberB. The best decision is to hire the candidate if and only ifA+B ≥ 100.

Alice and Bob sequentially report (with Alice speaking first) their observations to thecommittee chair, Charlie, who is to make the hiring decision. They report whether theirrespective observation is “high” or “low”.

Suppose A and B are uniformly and independently distributed on [0,100]. The beststrategy is the following: Alice reports “A is high” if A ≥ 50 and “A is low” otherwise. IfAlice reports “A is high”, then Bob reports “B is high” if B ≥ 25 and “B is low” otherwise;if Alice reports “A is low”, then Bob reports “B is high” if B ≥ 75 and “B is low” otherwise.Charlie hires the candidate if and only if Bob reports “B is high”.

It is important to discuss in greater detail how Example 4 fits the simple baseline modelwe introduced at the beginning of the section because this game will be used as the bench-mark model for our experiments that appear later. There are two types of equilibria inthis game. In the first type of equilibria, Alice babbles, i.e., Alice’s reports do not containany useful information about the candidate’s ability, the case in which context is not prop-erly defined. In the second type of equilibria, however, Alice’s reports are informative andthus provide the context in which Bob’s report is to be correctly interpreted. Conditionalon Alice’s report being informative, the later-stage extensive subgame that starts withBob’s decision node can be formalized with the state set Ω = F × C and the message setM = “B is high”, “B is low”, where C = “A is high”, “A is low” and F = [0,100].

The Efficiency Advantage of Literal Vagueness

We show that literally vague language may have an efficiency advantage over literallyprecise language. Therefore, the main result of Lipman (2009) is reverted as we broaden

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the notion of vagueness.

Proposition 1. Suppose ∣F ∣ ≥ 2, ∣C ∣ ≥ 2, ∣A∣ ≥ 2 and min∣F ∣∣C ∣−1, ∣A∣ ≥ ∣M ∣ ≥ 2. There existsa payoff function u ∶ Ω ×A → R under which the language in any optimal perfect Bayesianequilibrium must be literally vague.

Proof. Pick any payoff function u such that

U1. Every state has a unique optimal action.

U2. The set Tu ∶= a ∈ A ∶ a is the optimal action in some state ω ∈ Ω has ∣M ∣ elements.

U3. There is some a ∈ Tu such that the set Ω(a) ∶= ω ∈ Ω ∶ a is the optimal action in ω isnot the Cartesian product of a subset of F and a subset of C.

U1 and U2 imply that states can be partitioned into ∣M ∣ subsets such that states in thesame subset share the same unique optimal action and states in different subsets differin the optimal action. U3 implies that not every subset in this partition is the Cartesianproduct of a subset of F and a subset of C.

By U1 and U2, any optimal language is a bijection from Ω(a) ∶ a ∈ Tu to ∣M ∣. Thus,Πλ = Ω(a) ∶ a ∈ Tu. U3 and Lemma 1 thus imply that none of the optimal languages isliterally precise.

Remark 1. The conditions ∣F ∣ ≥ 2, ∣C ∣ ≥ 2 and ∣A∣ ≥ 2 reflect that the state set andthe decision problem are not trivial. ∣M ∣ ≥ 2 reflects that communication is not trivial.∣F ∣∣C ∣ − 1 ≥ ∣M ∣ implies that the perfect description of every state is impossible, that is, noone-to-one relation between states and messages exists. It is noteworthy that the wholeissue of vagueness would be less of a concern if messages were not limited because theperfect description is best whenever feasible.

Remark 2. ∣A∣ ≥ ∣M ∣ is not a necessary condition. The assumption is made to simplifythe proof.

Remark 3. Note that Proposition 1 is robust to the players’ prior beliefs because ex postoptimality is achieved in any optimal equilibrium under the payoff function we construct.It follows that a potential ex ante conflict of interest due to heterogeneous priors becomesirrelevant concerning the optimal equilibria because they are the most preferred strategyprofiles by both players regardless of priors. If we restrict the priors, for example, byassuming common prior, then the set of payoff functions under which Proposition 1 holdscould significantly expand. Therefore, for literal vagueness to be optimal, conditions U1-U3 are sufficient but not necessary.

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Remark 4. How common are payoff functions that satisfy the construction in the proofof Proposition 1? To get a sense of this answer, suppose ∣A∣ = ∣M ∣ = 2. It is clear that a payofffunction almost surely satisfies U1. Each payoff function u satisfying U1 induces a strictpreference ≻ωu such that a ≻ωu a

′ if and only if u(ω, a) > u(ω, a′) for any a, a′ ∈ A. Thereare a total of 2∣F ∣∣C∣ preference profiles inducible by a payoff function satisfying U1. Amongthese preference profiles, 2∣F ∣∣C∣−2 imply U2. Further, among these, 2∣F ∣+2∣C∣−4 violate U3.Therefore, the fraction of preferences satisfying U2 and U3 among those satisfying U1 7

is

2∣F ∣∣C∣ − 2∣F ∣ − 2∣C∣ + 2

2∣F ∣∣C∣.

This bound approaches 1 as ∣F ∣ and ∣C ∣ grow. Thus, for an environment with a large statespace, it is very likely that the optimal language is literally vague.

Remark 5. Note that Examples 2 and 3 do not immediately fit in the standard model,because in the standard model, unlike in Example 2, the message set does not vary withthe context, and unlike in Example 3, the listener does not receive a signal. Therefore,Proposition 1 does not immediately apply to these examples. It is nonetheless not difficultto tailor the model to these examples. In particular, Blume and Board (2013) presentresults in spirit similar to Proposition 1 for Example 2.

3 Experimental Implementation

We have seen that literally vague languages can be more efficient than literally preciselanguages in transmitting information. However, using a literally vague language may bechallenging in reality—it requires the speaker to adopt a more complicated strategy andthe listener to expect that correctly. If the strategic hurdle is so high as to overshadowthe efficiency advantage, then literal vagueness loses its appeal and is less likely to be asensible explanation for the prevalence of vague languages. Hence, we want to investigatewhether people can effectively use literally vague languages to communicate in reality.We design our experiments with this question in mind. Of course, a positive finding alonedoes not sufficiently prove that the prevalence of linguistic vagueness is founded on theefficiency advantage of literal vagueness in our sense, yet it is a worthwhile first step.

7We do not instead compute the fraction of preferences satisfying U1-U3 among all preferences becausepreferences with indifference compose a majority of all preferences when the number of alternatives is large,despite that utility functions with indifference are only measure zero.

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3.1 Experimental Games

We use the situation described in Example 4 to examine whether, and if so, how, peopleuse literally vague language in conversation. This example has a number of important andattractive features that make it particularly suitable for our purpose. First, as we showbelow, the optimal language, which is literally vague, has a moderate degree of efficiencyadvantage, which renders literal vagueness potentially useful but not entirely crucial forcommunication. Second, the situation is simple and straightforward, and thus, in princi-ple, it should shorten the learning period and make the experimental results more similarto the eventual stable language, or in terms of Lewis (1969), the “convention”. Third, thesituation covers the most general environment in which the context, that is, the messagefrom Alice, is endogenously generated. Fourth, and most important, this game has dif-ferent types of equilibria, one in which context is properly defined and another in whichcontext does not emerge endogenously. So, the emergence of a context-dependent, literallyvague language is not a necessary condition for an equilibrium. However, it is necessarythat individuals in this communication environment use a literally vague language in theoptimal equilibrium of the game.

The Benchmark Game

There are three players: two senders, Alice and Bob, and one receiver, Charlie. Aliceprivately observes a number A and Bob a number B. A and B are independently anduniformly drawn from [0,100]. Alice sends a message to Bob, where the message is either“A is Low” or “A is High”. Alice’s message is unobservable to Charlie. Then, Bob sendsa message to Charlie, where the message is either “B is Low” or “B is High”. Charliereceives Bob’s message (but not Alice’s message) and chooses an action: UP or DOWN.Players’ preferences are perfectly aligned. If A+B ≥ 100 and UP is chosen, or if A+B ≤ 100

and DOWN is chosen, then all receive a payoff of 1. Otherwise, all receive a payoff of 0.

Equilibria of the game can be classified into three categories:8

1. Bob babbling: Bob uses a strategy such that Charlie’s posterior about A+B remainsthe same as the prior upon seeing any message chosen by Bob with positive proba-bility. Whether Alice babbles or not does not bear on the outcome. No informationis transmitted, and Charlie is indifferent between the two actions regardless of themessage he receives. Accordingly, the success rate, which is the probability that

8A similar categorization of equilibria persists for variations of the benchmark game to be introduced.For those variations, we will skip the analysis of equilibria in which a converser babbles because they are ofno theoretical consequence and do not correspond to the experiment results.

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Charlie chooses the optimal action, is 50%.

2. Alice babbling: Bob sends message “B is High” if B > 50 or “B is Low” if B < 50.9Charlie chooses UP seeing “B is High” and DOWN otherwise. Only Bob’s informationis transmitted. Accordingly, the success rate is 75%.

3. No babbling: Alice sends “A is High” if A > 50 and “A is Low” if A < 50.

• If Alice’s message is “A is High”, then Bob sends “B is High” if B > 25 and “B isLow” if B < 25.

• If Alice’s message is “A is Low”, then Bob sends “B is High” if B > 75 and “B isLow” if B < 75.

Charlie chooses UP seeing “B is High” and DOWN otherwise. Accordingly, the suc-cess rate is 87.5%.

Clearly, any equilibrium in which no one babbles is optimal. Because the messagesavailable to Bob explicitly refer to the value of B alone, Bob’s language in any optimalequilibrium is clearly literally vague because the set of values of B that a particular mes-sage – say, “B is high” – describes depends on Alice’s message, which serves as the context.

To best test whether the players indeed effectively use the optimal literally vague lan-guage, we consider the counterfactual in which they do not, which can be formally modelledas Bob being constrained to use the same messaging strategy regardless of Alice’s message.In this case, the best Bob can do is to always use the cutoff of 50, and the correspondingsuccess rate is 75%. To examine the results and test whether the counterfactual holds, wepay particular attention to how Bob’s use of language depends on Alice’s message.

In addition, for a literally vague language to be effectively used, the listener shouldalso be aware of the underlying context dependence and correctly consider it when makingdecisions. However, Charlie’s strategy in the counterfactual would be the same as that inan optimal equilibrium so that considering only the benchmark will not allow us to identifywhether the listener fully understands the optimal, context-dependent language or not.Thus, we need further variations for sharper identification.

Variation 1 (Charlie hears Alice).

Consider a variation of the benchmark: the only difference is that Alice’s messages arenow also observable to Charlie. The equilibria in which no one babbles remain the same

9Of course, there are outcome-equivalent equilibria in which Bob uses the messages in the opposite way.We do not explicitly itemize such equilibria here and thereafter.

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as in the benchmark. It is particularly notable that Charlie’s strategy does not depend onAlice’s message even though it is observable.

In this variation, if Charlie believes that Bob uses his language in the optimal, literallyvague way, the former’s choice should not depend on Alice’s message because the informa-tion contained in Alice’s message is fully incorporated into Bob’s message through contextdependence. Thus, we can tease out whether Charlie correctly interprets Bob’s messageaccording to the optimal language by checking whether Charlie’s choice depends on Alice’smessage.

Variation 2 (Charlie chooses from three actions).

In this variation, Charlie has three actions to choose from: UP, MIDDLE and DOWN.UP is optimal if A +B ≥ 120, MIDDLE if 80 ≤ A +B ≤ 120, and DOWN if A +B ≤ 80. If theoptimal action is chosen, the players all receive a payoff of 1 and 0 otherwise.

In equilibria in which no one babbles, Alice sends “A is High” if A > 50 and “A is Low”if A < 50.

• If Alice’s message is “A is High”, then Bob sends “B is High” if B > 25 and “B is Low”if B < 25.

• If Alice’s message is “A is Low”, then Bob sends “B is High” if B > 75 and “B is Low”if B < 75.

Charlie chooses UP seeing “B is High” and DOWN otherwise. Accordingly, the successrate is 63.5%. It should be noted that MIDDLE is never chosen in this equilibrium.

The purpose of introducing the third action is to make the conversational environmentmore complex, particularly for Bob and Charlie. The benchmark and variation 1 are rela-tively simple environments in which it is not supremely difficult to “compute” the optimalcutoffs.10 However, when people talk in real life, they do not typically derive the opti-mal language consciously. Thus, we aim to determine whether people can still arrive atthe optimal, literally vague language and use it to effectively communicate when it ismore difficult to explicitly derive the optimal language or whether they instead revert tocontext-independent language, which is simpler for the speaker to use and for the listenerto understand. Thus, it is crucial to examine whether or not Bob’s message is contextdependent in this variation and whether Charlie best responds or not.

Variation 3 (Charlie chooses from three actions and hears Alice).10The key logic is that Alice would use a cutoff of 50 because of the problem’s symmetry. Thereafter, the

optimal cutoffs of 25 and 75 can be deduced in one’s head or at most in a back-of-envelope calculation.

14

This variation differs from Variation 2 in that Alice’s message is now observable toCharlie. In an optimal equilibrium, Alice sends “A is High” if A > 50 and “A is Low”if A < 50. Bob’s strategy does not differ from that in Variation 2 qualitatively but hasdifferent optimal cutoffs.

• If Alice’s message is “A is High”, then Bob sends “B is High” if B > 45 and “B is Low”if B < 45.

• If Alice’s message is “A is Low”, then Bob sends “B is High” if B > 55 and “B is Low”if B < 55.

Charlie chooses UP seeing (“A is High”, “B is High”), DOWN seeing (“A is Low”, “B isLow”), and MIDDLE otherwise. The success rate is 78.5%. It should be noted that MID-DLE is chosen in this equilibrium when the messages from Alice and Bob disagree.

This variation serves two purposes. First, it allows us to study how the quantitativechange in the optimal cutoff values affects language use. The optimal cutoff values thatare substantially closer to one another generates a very minimal benefit of context depen-dence relative to the context-independent counterfactual. In fact, the success rate fromthe context-independent counterfactual is 78%. Thus, this variation enables us to under-stand how individuals’ choice of context-dependent languages is guided by the salience ofincentives.

Variation 4 (Bob’s Messages are Imperative).

Consider a variation of the Benchmark in which we replace Bob’s messages “B is High”and “B is Low” by “Take UP” and “Take DOWN”. Clearly, this change eradicates any pos-sibility of literal vagueness because Bob’s messages, now imperative, have unambiguousliteral interpretations with respect to the decision problem at hand. Apart from the differ-ence in the literal interpretation of messages, the variation is the same as the benchmark.Hence, the variation serves as a useful control version of the benchmark. In particular, itallows us to test whether literal vagueness may intimidate players from using the optimallanguage.

In the experiments, we create not only an “imperative messages” version of the bench-mark but also one of variation 2.

3.2 Experimental Design and Hypotheses

The benchmark game and its variants introduced in the previous section constitute ourexperimental treatments. Our experiment features a (2×2)+(2×1) treatment design (Table

15

1). The first treatment variable concerns the number of actions available to the receiver(Charlie), and the second treatment variable concerns whether or not Alice’s messagesare observed by Charlie. The third treatment variable concerns whether Bob’s messagesare framed to be indicative or imperative.11 We consider the treatments with imperativemessages as a robustness check and therefore omit the corresponding treatments in whichAlice’s messages are observed by Charlie.

Table 1: Experimental Treatments

Indicative Messages from Bob

Alice’s messages# of Actions

Two ThreeUnobservable 2A-U-IND 3A-U-INDObservable 2A-O-IND 3A-O-IND

+Imperative Messages from Bob

Alice’s messages# of Actions

Two ThreeUnobservable 2A-U-IMP 3A-U-IMP

Note that we design our experiments to obtain empirical evidence on the efficiency foun-dation of literal vagueness. Hence, all our (null) hypotheses in this section are stated basedon the assumption that the optimal equilibrium will be achieved in the laboratory. We payattention to several properties of the optimal equilibrium (e.g., whether the observabilityof Alice’s message to Charlie affects the success rate) and comparative statistics that re-sult from the optimal equilibrium. Those properties do not necessarily hold in some other,Pareto-suboptimal equilibria.

Our first experimental hypothesis concerns the overall outcome of the communicationgames represented by the success rate. Let S(T ) denote the average success rate of Treat-ment T . There are two qualitative characteristics predicted by the optimal equilibriumwith context-dependent, literally vague language. First, the observability of Alice’s mes-sage to Charlie does not affect the success rate in the treatments with two actions, whilethe same observability increases the success rate in the treatments with three actions.Second, the imperativeness of Bob’s messages does not affect the success rate. Thus, wehave the following hypothesis.

Hypothesis 1 (Success Rate).

1. Observability of Alice’s message to Charlie does not affect the success rate in thetreatments with two actions, i.e., S(2A-O-IND) = S(2A-U-IND).

2. Observability increases the success rate in the treatments with three actions, i.e.,S(3A-O-IND) > S(3A-U-IND).

11For example, Bob’s message spaces in Treatments 2A-U-IND and 2A-U-IMP are “B is HIGH”, “B isLOW” and “Take UP”, “Take DOWN”, respectively.

16

3. The imperativeness of Bob’s messages does not affect the success rate, i.e., S(2A-U-IND) = S(2A-U-IMP) and S(3A-U-IND) = S(3A-U-IMP).

Our second hypothesis considers the counterfactual in which Bob is constrained to becontext independent and thus always uses the cutoff of 50.12 In the counterfactual sce-nario, the success rates are 75% in Treatments 2A-O-IND, 2A-U-IND, and 2A-U-IMP,78% in Treatment 3A-O-IND, and 55% in Treatments 3A-U-IND and 3A-U-IMP. If theplayers effectively use the optimal, literally vague language, the success rates should besignificantly above the levels predicted by the counterfactual. Thus, we have the followinghypothesis.

Hypothesis 2 (Counterfactual Comparison). The observed success rate in each treatmentis significantly higher than the success rate calculated based on the counterfactual in whichBob is constrained to be context independent. More precisely,

1. S(2A-O-IND), S(2A-U-IND), S(2A-U-IMP) > 75%

2. S(3A-O-IND) > 78%

3. S(3A-U-IND), S(3A-U-IMP) > 55%

Note that the success rate predicted by the optimal, literally vague language in Treat-ment 3A-O-IND is 78.5%, which is not substantially different from the 78% predicted bythe counterfactual. The net benefit of the context-dependent language measured with re-spect to the success rates is only 0.5% (= 78.5% - 78%) in Treatment 3A-O-IND. The netbenefit of context dependence becomes substantially larger in other treatments as it is12.5% (= 87.5% - 75%) in Treatments 2A-O-IND, 2A-U-IND, and 2A-U-IMP, and 6.5% (=63.5% - 55%) in Treatments 3A-U-IND and 3A-U-IMP.

Our third hypothesis concerns Alice’s message choices. For all treatments, the optimalequilibrium play predicts that Alice employs the simple cutoff strategy in which she sends“A is High” ifA > 50 and “A is Low” ifA < 50. Let PAlice(m∣A) denote the proportion of Alice’smessage m given the realized number A. Then PAlice(“A is Low”∣A) = 1 for any A < 50 andPAlice(“A is High”∣A) = 1 for any A > 50.

Hypothesis 3 (Alice’s Messages). Alice’s message choices observed in all treatments arethe same. Moreover, Alice employs the simple cutoff strategy in which she sends “A is High”if A > 50 and “A is Low” if A < 50.

12Charlie’s optimal strategy remains the same regardless of whether Bob is constrained to be contextindependent or not.

17

Our next hypothesis concerns Bob’s message choices. The optimal equilibrium play pre-dicts that Bob’s message choices depend on the context, i.e., which message he receivedfrom Alice. To state our hypothesis clearly, let Pm′

Bob(m∣B) denote the proportion of Bob’smessagem given the realized numberB and Alice’s messagem′ ∈ “A is High”, “A is Low”.Define

CD(B) = PLBob(m∣B) − PH

Bob(m∣B)

where m is “B is Low” for the treatments with indicative messages and “Take DOWN” forthe treatments with imperative messages. CD(B) measures the degree of context de-pendence of Bob’s message choices given the realized number B. Bob’s optimal, context-dependent strategy implies that there is a range of number B under which Bob’s messagechoices differ depending on Alice’s messages, i.e., CD(B) = 1. Such intervals are [45,55]

for Treatment 3A-O-IND and [25,75] for all other treatments. It is worth noting thatCD(B) = 0 for any B ∈ [0,100] if Bob uses a context-independent strategy. Thus, we havethe following hypothesis:

Hypothesis 4 (Bob’s Messages). Bob’s messages are context dependent in a way that ispredicted by the optimal equilibrium of each game. More precisely,

1. For each treatment T , there exists an interval [XT , Y T ] with XT > 0 and Y T < 100 suchthat CD(B) > 0 for any B ∈ [XT , Y T ] and CD(B) = 0 otherwise.

2. The length of the interval [XT , Y T ] is significantly smaller in Treatment 3A-O-INDthan in any other treatment.

Our last hypothesis concerns whether the listener, Charlie, can correctly interpret themessages. In particular, Charlie’s strategy in the optimal equilibrium does not dependon whether or not Alice’s messages are observable to Charlie in the treatments with twoactions. In the games with three actions, however, the observability of Alice’s message toCharlie matters. Precisely, the optimal equilibrium predicts that MIDDLE should not betaken by Charlie in Treatments 3A-U-IND and 3A-U-IMP, whereas MIDDLE is taken inTreatment 3A-O-IND when the messages from Alice and Bob do not coincide. Thus, wehave the following hypothesis.

Hypothesis 5 (Charlie’s Action Choices). In the treatments with two actions, Charlie’saction choices do not depend on whether Alice’s messages are observable. In the treatmentswith three actions, MIDDLE is taken only in Treatment 3A-O-IND when the messages fromAlice and Bob do not coincide.

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3.3 Procedures

Our experiment was conducted in English using z-Tree (Fischbacher (2007)) at the HongKong University of Science and Technology. A total of 138 subjects who had no prior ex-perience with our experiment were recruited from the undergraduate population of theuniversity. Upon arrival at the laboratory, subjects were instructed to sit at separatecomputer terminals. Each received a copy of the experimental instructions. To ensurethat the information contained in the instructions was induced as public knowledge, theinstructions were read aloud, aided by slide illustrations and a comprehension quiz.

We conducted a session for each treatment. In all sessions, subjects participated in21 rounds of play under one treatment condition. Each session had 21 or 24 participantsand thus involved 7 or 8 fixed matching groups of three subjects: a Member A (Alice), aMember B (Bob), and a Member C (Charlie). Thus, we used the fixed-matching protocoland between-subjects design. As we regard each group in each session as an indepen-dent observation, we have seven to eight observations for each of these treatments, whichprovide us with sufficient power for non-parametric tests. At the beginning of a session,one-third of the subjects were randomly labeled as Member A, another third labeled asMember B and the remaining third labeled as Member C. The role designation remainedfixed throughout the session.

We illustrate the instructions for Treatment 2A-U-IND. The full instructions can befound in Appendix A. For each group, the computer selected two integer numbers A andB between 0 and 100 (uniformly) randomly and independently. Subjects were presentedwith a two-dimensional coordinate system (with A in the horizontal coordinate and B inthe vertical coordinate) as in Figures 5(a) and 5(b) in Appendix A. The selected number Awas revealed only to Member A, and the selected number B was revealed only to MemberB. Member A sent one of two messages – “A is Low” or “A is High” – to Member B but notto Member C. After observing both the selected number B and the message from MemberA, Member B sent one of two messages – “B is Low” or “B is High” – to Member C, whothen took one of two actions: UP or DOWN. The ideal actions for all three players wereUP when A +B > 100 and DOWN when A +B < 100.13 Every member in a group received50 ECU if the ideal action was taken and 0 ECU otherwise.

For Rounds 1-20, we used the standard choice method so that each participant firstencountered a possible contingency and specified a choice for the given contingency. Forexample, Member A decided what message to send after seeing the randomly selectednumberA. Member B decided what message to send after observing the randomly selected

13To make the likelihood of each action being ideal exactly equal across two actions, we set both actions tobe ideal when A +B = 100.

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number B and the message from Member A. Similarly, Member C decided what action totake after receiving the message from Member B. For Round 21, however, we used thestrategy method and elicited players’ beliefs by providing a small amount of compensation(in the range between 2 ECU and 8 ECU) for each correct guess.14 For more details, seethe selected sample scripts for the strategy method and the belief elicitation provided inAppendix B.

We randomly selected two rounds of the 21 total rounds for each subject’s payment.The sum of the payoffs a subject earned in the two selected rounds was converted intoHong Kong dollars at a fixed and known exchange rate of HK$1 per 1 ECU. In additionto these earnings, subjects also received a payment of HK$30 for showing up. Subjects’total earnings averaged HK$103.5 (≈ US$13.3).15 The average duration of a session wasapproximately 1 hour.

4 Experimental Findings

4.1 Overall Outcome

Note: The red bars depict the theoretical predictions from the optimal,literally vague equilibria. The red dotted lines depict the predictions fromthe counterfactual in which Bob is constrained to be context independent.

Figure 1: Average Success Rate

We report our experimental results as a number of findings that address our hypothesesas shown in Section 3.2.

14Although we were aware that an appropriate incentive-compatible mechanism is needed to correctlyelicit beliefs, we used this elicitation procedure because of its simplicity and the fact that the belief andstrategy data were only secondary data used mainly for the robustness checks.

15Under the Hong Kong currency board system, the Hong Kong dollar is pegged to the US dollar at therate of HK $7.8 = US$1.

20

Figure 1 reports the average success rates aggregated across all rounds and all match-ing groups for each treatment and presents the theoretical predictions from the optimalequilibrium depicted by the red bars and the predictions from the counterfactual in whichBob is constrained to be context independent, as depicted by the dotted lines. Severalobservations were apparent. First, a non-parametric Mann-Whitney test reveals that thesuccess rates in Treatment 2A-O-IND and in Treatment 2A-U-IND were not statisticallydifferent (81.6% vs. 83.6%, two-sided, p-value = 0.6973). In contrast, the success rate inTreatment 3A-O-IND was 73.8%, which is significantly higher than the 54.2% in Treat-ment 3A-U-IND (Mann-Whitney test, p-value = 0.0267). This observation is consistentwith Hypothesis 1 that the observability of Alice’s message to Charlie affects the successrate only in the treatments with three actions.

Second, there was no significant difference in the success rates between Treatment 2A-U-IND and Treatment 2A-U-IMP (83.6% vs. 85.7%, two-sided Mann-Whitney test, p-value= 0.5989) and between Treatment 3A-U-IND and Treatment 3A-U-IMP (54.2% vs. 51.2%,two-sided Mann-Whitney test, p-value = 0.2237). This observation is also consistent withHypothesis 1 that the imperativeness of Bob’s message does not affect the success rate,regardless of the number of available actions. Confirming Hypothesis 1, we thus have ourfirst finding as follows:

Findings 1. Observability of Alice’s message to Charlie affected the success rate only inthe treatments with three actions. The imperativeness of Bob’s message did not affect thesuccess rate regardless of the number of available actions.

Figure 1 seems to suggest that the success rates observed in the three treatments withtwo actions (hereafter, Treatments 2A) are better approximated by the predictions fromthe optimal, context-dependent equilibrium languages than by the predictions from thecontext-independent counterfactual. Indeed, we cannot reject the null hypothesis that thesuccess rates are not different from 87.5%, the predicted value from the optimal equilib-rium (two-sided Mann-Whitney tests, p-values are 0.8262, 0.5076, 1.000 for Treatments2A-O-IND, 2A-U-IND and 2A-U-IMP, respectively). Even if we can reject the alternativehypothesis that the success rates are significantly higher than the predicted level of 75%from the context-independent counterfactual only for Treatment 2A-U-IND (one-sidedMann-Whitney tests, p-values are 0.2551, 0.0610, 0.2174 for Treatments 2A-O-IND, 2A-U-IND and 2A-U-IMP, respectively), the p-values that resulted from the non-parametricanalysis suggest that the optimal, context-dependent equilibrium is a better predictor ofthe results observed from these treatments.

On the other hand, we do not receive the same observation from the three treatmentswith three actions (hereafter, Treatments 3A), especially those with unobservable mes-

21

sages from Alice. We cannot reject the null hypothesis that the success rates observed inthese treatments are the same as the success rates predicted by the context-independentcounterfactual (two-sided Mann-Whitney tests, p-values are 0.4347, 0.6936, and 0.4347,respectively, for Treatments 3A-O-IND, 3A-U-IND and 3A-U-IMP). Moreover, the successrates observed in Treatments 3A-U-IND and 3A-U-IMP were 54.2% and 51.2%, respec-tively, which are substantially lower than the predicted level of 63.5% from the optimal,literally vague equilibrium language. Although the difference is statistically insignificant(two-sided Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively), the p-valuesgenerated from the non-parametric analysis suggest that the context-independent coun-terfactual is a better predictor of the results from Treatments 3A-U-IND and 3A-U-IMP.16

Thus, we have the following result:

Findings 2. The average success rates observed in Treatments 2A-O-IND, 2A-U-IND, and2A-U-IMP were higher than the predicted level from the counterfactual in which Bob iscontext independent. The average success rates observed in Treatments 3A-O-IND, 3A-U-IND, and 3A-U-IMP were lower than the predicted level from the counterfactual. However,the difference between the observed success rate and the prediction from the counterfactualis statistically significant only in Treatment 2A-U-IND.

Among Treatments 3A, more substantial deviations from the optimal, context-dependentequilibrium were observed in Treatments 3A-U-IND and 3A-U-IMP. This observed de-viation in the average success rates may imply that context-dependent, literally vaguelanguages did not emerge in those treatments, most likely due to the complexity of theenvironment considered. However, another completely plausible scenario would be thatthe observed deviation originates from a different source, such as Charlie’s choices beinginconsistent with the optimal equilibrium. Without taking a careful look at the individualbehaviour, it is impossible to draw any meaningful conclusions. Hence, in the subsequentsections, we look at individual players’ choices.

4.2 Alice’s Behaviour

Figure 2 reports Alice’s message strategies by presenting the proportion of each messageas a function of the number A, where data were grouped into bins by the realization ofnumberA (e.g., [0,5), [5,10), ..., etc.). Figure 2(a) provides the aggregated data from Treat-ments 2A, while Figure 2(b) provides the aggregated data from Treatments 3A. The samefigures separately drawn for each treatment can be found in Appendix C.

16The success rates observed in Treatments 3A-O-IND was 73.8%, which is not significantly different fromthe predicted level of 78.5% from the optimal, literally vague equilibrium language (two-sided Mann-Whitneytest, p-value = 0.4347).

22

0

20

40

60

80

100

[0,5)

[5,10)

[10,15)

[15,20)

[20,25)

[25,30)

[30,35)

[35,40)

[40,45)

[45,50)

[50,55)

[55,60)

[60,65)

[65,70)

[70,75)

[75,80)

[80,85)

[85,90)

[90.95)

[95,100]

Prop

or%on

Number A

Alice's Message -­‐ Treatments 2A

LOW

HIGH

(a) Treatments 2A

0

20

40

60

80

100

[0,5)

[5,10)

[10,15)

[15,20)

[20,25)

[25,30)

[30,35)

[35,40)

[40,45)

[45,50)

[50,55)

[55,60)

[60,65)

[65,70)

[70,75)

[75,80)

[80,85)

[85,90)

[90.95)

[95,100]

Prop

or%on

Number A

Alice's Message -­‐ Treatments 3A

LOW

HIGH

(b) Treatments 3A

Note: The red dotted lines illustrate the optimal equilibrium strategy with a cutoff of 50.

Figure 2: Alice’s Messages

From these two figures, it was immediately clear that the subjects with the designatedrole of Alice in our experiments tended to use cut-off strategies approximated well by theoptimal cutoff of 50. Using the matching-group-level data from all rounds for each binof the realized number A (e.g. [0,5), [5,10), ..., etc.) as independent data points for eachtreatment, a set of (two-sided) Mann-Whitney tests reveals that 1) for any bins of A below50, we cannot reject the null hypothesis that the proportion of message “A is Low” beingsent was 100%, and 2) for any bins of A above 50, we cannot reject the null hypothesis thatthe proportion of message “A is Low” being sent was 0%. Among 20 bins in each treatment,the p-values for 14-17 bins were 1.0, while the lowest p-value for each treatment rangedbetween 0.2636 and 0.5637. Confirming our Hypothesis 3, we have the following result:

Findings 3. For any treatment, PAlice(“Low”∣A) = 1 for any A < 50 and PAlice(“High”∣A) = 1

for any A > 50.

The elicited strategies and beliefs reported in Figure 10 in Appendix C provided ad-ditional support for Finding 3. We cannot reject the null hypothesis that the reportedcutoff values for Alice’s strategy and other players’ reported beliefs for Alice’s strategy inall treatments were the same as the optimal equilibrium cutoff of 50 (two-sided Mann-Whitney tests, p-values range between 0.4533 and 1.000).

4.3 Bob’s Behaviour

We now examine Bob’s behaviour. Recall that PLBob(m∣B) denotes the proportion of Bob’s

message m given the realized number B and Alice’s message “A is Low”, and PHBob(m∣B)

denotes the proportion of Bob’s message m given Alice’s message “A is High”. We intro-

23

(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines present the predicted distribution from the optimal equilibrium strategy.

Figure 3: Bob’s Message Strategy

duced the measure of context dependence as CD(B) = PLBob(m∣B) − PH

Bob(m∣B) where m is“B is Low” for the treatments with indicative messages and “Take DOWN” for the treat-ments with imperative messages. Figures 3(a)-(f) illustrate the distributions of CD(B)

over the realization of number B, aggregated across all matching groups for each treat-ment. Again, the data from all rounds were grouped into bins by the realization of numberB (e.g., [0,5), [5,10), ..., etc.).

The optimal, context-dependent equilibrium language implies that there exists an in-

24

terval (strictly interior of the support of B) such that the value of CD(B) is 1 if B is inthe interval and 0 otherwise. Moreover, the boundaries of such intervals are determinedby the equilibrium cutoff strategy so that the interval is narrower in Treatment 3A-O-IND. The exact prediction of the distribution of CD(B) made by the optimal equilibriumis illustrated by the red-dotted lines in Figures 3(a)-(f).17

These figures convincingly visualize the fact that, in each treatment, there exists aninterval in which the value of CD(B) is strictly positive. For Treatment 2A-O-IND, forinstance, we cannot reject the null hypothesis that CD(B) = 0 for B ∈ [0,30) and for B ∈

[60,100] (two-sided Mann-Whitney test, both p-values = 1.00). However, for B ∈ [30,60), wecan reject the null hypothesis that CD(B) = 0 in favour of the alternative that CD(B) > 0

(one-sided Mann-Whitney test, p-value = 0.04).18 Similarly, for Treatments 2A-U-IMP and3A-U-IMP, we cannot reject the null hypothesis that CD(B) = 0 for the intervals of [25,75)

(p-values are 0.052 and 0.059, respectively) in favour of the alternative that CD(B) > 0.Qualitatively similar but less significant results were obtained from Treatment 2A-U-INDwith the non-zero interval of [25,70) (p-value = 0.136) and from Treatment 3A-U-IND withthe non-zero interval of [25,75) (p-value = 0.121). Thus, we have the following result:

Findings 4. In Treatments 2A-O-IND, 2A-U-IND, 2A-U-IMP, 3A-U-IND, and 3A-U-IMP,there existed an interval [X,Y ] with X > 0 and Y < 100 such that CD(B) > 0 if B ∈ [X,Y ]

and CD(B) = 0 otherwise.

We must discuss the data from Treatment 3A-O-IND in Figure 3(b) more carefully.First, as predicted by the optimal equilibrium strategy, the interval that has a non-zerovalue of CD(B) seemed to shrink significantly compared to any other treatments. Indeed,we cannot reject the null hypothesis that CD(B) = 0 for B ∈ [0,45) and for B ∈ [50,100]

(two-sided Mann-Whitney test, p-values are 1.00 and 0.7237, respectively). However, asubstantial deviation from the theoretical prediction was observed such that the reportedvalue for the bin [50,55) was negative.19 This observation was driven by the fact that theobserved cutoff value from Bob’s strategy conditional on Alice’s message “A is Low” was50, which may appear more focal than the theoretically optimal cutoff of 45.20

17Figures 8(a)-(f) and 9(a)-(f) presented in Appendix C separately report the distributions ofPLBob(“B is Low”∣B) and of PH

Bob(“B is Low”∣B) for each treatment.18To conduct Mann-Whitney tests for Hypothesis 4, we first eyeball Figures 3(a)-(f) to identify the plausible

choices of the interval with CD(B) > 0 for each treatment. For example, for Treatment 2A-O-IND, relyingon Figure 3(a), we divide the support of number B into three intervals – [0,30), [30,60), and [60,100]. Wenext calculate the value of CD(B) for each of the three intervals for each matching group. Taking thosevalues as group-level independent data points for each treatment, we conducted the non-parametric test.

19For Treatment 3A-O-IND, we cannot conduct any meaningful statistical analysis for B ∈ [45,50) becausethere are only two group-level independent data points.

20Similarly, two substantial deviations were observed in Treatment 3A-U-IND - in the first bin of [0,5)and the ninth bin of [40,45) in Figure 3(d). The first deviation was driven solely by the single data point

25

The elicited strategies and beliefs reported in Figure 11 in Appendix C provide furthersupporting evidence. Wilcoxon signed-rank tests reveal that the reported cutoff valuesgiven Alice’s message “A is Low” were significantly higher than the cutoff values givenAlice’s message “A is High” for most of the treatments (p-values ranged between 0.0004and 0.0204) except for Treatment 3A-O-IND and Treatment 3A-U-IND.21 The fact thatthe reported cutoff values in Treatment 3A-O-IND were not significantly different (two-sided, p-value = 0.1924) is not surprising at all because the optimal equilibrium cutoffvalues are 45 and 55, distinctively more similar to each other than the predicted valuesfor all other treatments. The insignificant result for Treatment 3A-U-IND (two-sided, p-value = 0.3561) originated mainly from two observations that the reported cutoffs fromtwo Charlie subjects were 50 and 45 given “A is Low” and 85 and 80 given “A is High”.

4.4 Charlie’s Behaviour

Figure 4 reports Charlie’s action choices by presenting the proportion of each action asa function of information available to Charlie. Figure 4(a) presents the data aggregatedacross all matching groups of Treatments 2A, while Figure 4(b) presents the data aggre-gated across all matching groups of Treatments 3A.

(a) Treatments 2A (b) Treatments 3A

Note: The red bars present the theoretical predictions from the optimal equilibria.

Figure 4: Charlie’s Actions

A few observations emerged immediately from these figures. First, Figure 4(a) revealsthat observed strategy by Charlie depended to a limited extent on whether Alice’s mes-with B ∈ [0,5) in which Bob sent “B is High” after receiving “A is High” from Alice. The second deviationwas driven solely by the single data point with B ∈ [40,45) in which Bob sent “B is Low” after receiving “Ais High” from Alice.

21To conduct Wilcoxon signed-rank tests, we pooled the data from the reported cutoff values for Bob’sstrategy and other players’ reported beliefs.

26

sages were observable to him in Treatments 2A. In Treatment 2A-O-IND, the proportionof UP that was chosen given Bob’s message “B is High” was about 52% and the proportionof DOWN that was chosen given Bob’s message “B is Low” was about 75%; both findingsare substantially and significantly different from 100% predicted by the optimal equilib-rium. However, the observed proportions become 63% and 100% if we take the last threerounds of data only, showing that learning took place in the right direction.22

Second, Figure 4(b) reveals that MIDDLE was taken by Charlie in Treatment 3A-O-IND when the messages from Alice and Bob did not coincide. The proportions of MIDDLEthat were chosen by Charlie given the message combinations (“A is High”, “B is Low”) and(“A is Low”, “B is High”) were higher than 90%. However, inconsistent with the predictionfrom the optimal equilibrium strategy, MIDDLE was taken even in Treatments 3A-U-INDand 3A-U-IMP. The proportions of MIDDLE that were taken in these two treatmentsvaried between 24% and 43%, which are substantially larger than 0%. This observeddeviation seemed persistent because it did not disappear even when we took the datafrom the last three rounds only.23 Thus, we have the following result:

Findings 5. Charlie’s observed action choices in Treatment 2A-O-IND were not the same asthose observed in Treatment 2A-U-IND, showing that the observability of Alice’s message toCharlie mattered. For Treatments 3A, MIDDLE was taken in Treatment 3A-O-IND whenCharlie received different messages from Alice and Bob. However, a substantial proportionof MIDDLE was observed even in Treatments 3A-U-IND and 3A-U-IMP.

This observed discrepancy in Charlie’s action choices between our data and the pre-diction from the optimal equilibrium was the main source of the lower success rates wehad in Treatments 3A-U-IND and 3A-U-IMP (see Figure 1). The success rates observedin Treatments 3A-U-IND and 3A-U-IMP were 54.2% and 51.2%, respectively, which aresubstantially lower than the predicted level of 63.5%, although the difference is statisti-cally insignificant (Mann-Whitney tests, p-values are 0.4308 and 0.2413, respectively). Ifwe replace Charlie’s observed empirical choices with the hypothetical choices from the op-timal strategy, the success rates in Treatments 3A-U-IND and 3A-U-IMP become 58.3%and 56.0%, respectively; both are substantially higher than the observed levels.

Note that MIDDLE was taken slightly more often in Treatment 3A-U-IMP than inTreatment 3A-U-IND (31% and 24% vs. 33% and 43%). The elicited strategies and beliefs

22In an early stage of the project, we conducted two sessions in which subjects participated in the first 20rounds with the treatment condition of 2A-U-IND and in the second 20 rounds with the treatment conditionof 2A-O-IND. The data from the second 20 rounds were almost perfectly consistent with the theoreticalprediction, showing convincing evidence of learning. Data from this additional treatment are availableupon request.

23The elicited strategies and beliefs presented in Figure 12 in Appendix C were highly consistent with theresults in Finding 5.

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presented in Figure 12 in Appendix C demonstrate this difference more vividly. This dif-ference may originate from the fact that we framed Bob’s messages as “Don’t take UP” and“Don’t take DOWN” to impose imperativeness to the messages for Treatment 3A-U-IMP.

4.5 Emergence of Context-dependent, Literally Vague Languages

In this section, we combine our findings presented in the previous sections to establishthe emergence of context-dependent, literally vague languages. Admittedly, we did notpresent a perfect match between our data and the prediction from the optimal, context-dependent equilibrium language. However, we provided convincing evidence that overallbehaviour observed in our laboratory was qualitatively consistent with the prediction.More precisely,

1. Finding 3 illustrated that Alice tended to use cut-off strategies approximated well bythe cut-off value of 50. Thus, the contexts are properly defined.

2. Finding 4 showed that Bob tended to use context-dependent strategies.

3. Finding 5 suggested that Charlie understood the messages from the speaker(s) welland took actions in a manner that is qualitatively consistent with the optimal equi-librium strategy.

4. Finding 5 also revealed that the lower success rates observed in Treatments 3A-U-IND and 3A-U-IMP reported in Finding 2 were driven largely by Charlie’s behaviourbeing partially inconsistent with the theoretical predictions from the optimal equi-librium.

Taking these findings together, we establish the following result:

Findings 6. Literally vague language emerged in our experiment, and the ways it was usedby the speakers and understood by the listener were all consistent with the prediction fromthe optimal equilibria of the communication games.

5 Related Literature

Economists have studied strategic language use since the cheap talk model was proposedby Crawford and Sobel (1982). Despite the formidable literature since generated on thissubject, only recently was linguistic vagueness given the academic attention it deserveswhen Lipman (2009), asking “why is language vague”, defined vagueness and showed that

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vagueness has no advantage in the standard cheap talk setting with a common interest. Inthis line, Blume and Board (2013) explore a situation in which the linguistic competenceof some conversing parties is unknown and show that this uncertainty could make theoptimal language vague. The same authors further investigate the effect of higher-orderuncertainty about linguistic ability on communication in Blume and Board (2014a) andfind that in the common interest case, vagueness persists but efficiency loss due to higher-order uncertainty is small. Lambie-Hanson and Parameswaran (2015) study a model inwhich the listener may have a limited ability to interpret messages, and the sender mayor may not be aware of this. The authors find that the optimal language can be vague.In Section 2, particularly by Examples 2 and 3, we show that these models that enrichthe simplest cheap talk can be unified and incorporated in our model by taking the meta-conversational uncertainty as the context dimension of the state. Our Proposition 1, whiletechnically different from their findings, resonates the same keynote.

It is well known that even in the presence of a conflict of interest, endogenous vaguelanguage still has no efficiency advantage in the cheap talk framework. However, Blume,Board, and Kawamura (2007) show that exogenous noise in communication, which forcesvagueness upon the language, can generate a Pareto improvement. Blume and Board(2014b) further confirm that the speaker may intentionally exploit the noise to introducemore vagueness in the language. These papers differ from ours in that we focus on thecommon interest environment, and moreover, we study the efficiency foundation of en-dogenous vagueness.

Outside the field of economics, Lewis (1969) is a seminal philosophic analysis of lan-guage on this subject. Keefe and Smith (1997) and Keefe (2007) present extensive philo-sophical discussion on vague language, particularly vague language that gives rise to theSorites paradox. van Deemter (2010) finds that in the field of computer science, vagueinstruction may make a search process more efficient than precise instruction.

On the experimental side, several recent papers investigate how the availability ofvague messages improves or preserves efficiency. Serra-Garcia, van Damme, and Pot-ters (2011) show that vague language helps players preserve efficiency in a two-playersequential-move public goods game with asymmetric information. Wood (2016) exploresthe efficiency-enhancing role of vague language in a discretized version of canonical sender-receiver games a la Crawford and Sobel (1982). Agranov and Schotter (2012) show thatvague messages are useful in concealing conflict between a sender and a receiver that,if it is publicly known, would prevent the parties from coordinating and achieving an ef-ficient outcome. All these papers take the availability of vague messages as given andstudy how it affects players’ coordination behaviour. In contrast, we explore how a mes-sage endogenously obtains its vague meaning. For a more comprehensive discussion of

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the experimental literature on vague language, see the recent survey by Blume, Lai, andLim (2017).

6 Concluding Remarks

In this paper, we introduce the notion of context-dependence and literal vagueness, andoffer our explanation of why language is vague. We show that literal vagueness arisesin the optimal equilibria in many standard conversational situations. Our experimentaldata provide supporting evidence for the emergence of literally vague languages.

Although our discussion of linguistic vagueness focuses on an environment in whichplayers’ preferences are perfectly aligned, the theoretical discussions presented in Blume,Board, and Kawamura (2007) and Blume and Board (2014b) suggest that the communica-tive advantage of literal vagueness would be extended to an environment with conflicts ofinterest. We believe that experimentally investigating the role of vagueness in the pres-ence of conflicts of interest is an interesting avenue for future research.

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ReferencesAgranov, Marina and Andrew Schotter (2012), “Ignorance is bliss: An experimental study

of the use of ambiguity and vagueness in the coordination games with asymmetric pay-offs.” American Economic Journal: Microeconomics, 4, 77–103.

Blume, Andreas and Oliver Board (2013), “Language barriers.” Econometrica, 81, 781–812.

Blume, Andreas and Oliver Board (2014a), “Higher-order uncertainty about language.”Working paper.

Blume, Andreas and Oliver Board (2014b), “Intentional vagueness.” Erkenntnis, 79, 855–899.

Blume, Andreas, Oliver J. Board, and Kohei Kawamura (2007), “Noisy talk.” TheoreticalEconomics, 2, 395–440.

Blume, Andreas, Ernest K. Lai, and Wooyoung Lim (2017), “Strategic information trans-mission: A survey of experiments and theoretical foundations.” Working paper.

Crawford, Vincent and Joel Sobel (1982), “Strategic information transmission.” Economet-rica, 50, 1431–51.

Cremer, Jacques, Luis Garicano, and Andrea Prat (2007), “Language and the theory ofthe firm.” The Quarterly Journal of Economics, 122, 373–407.

Dilme, Francesc (2017), “Optimal languages.” Working paper.

Fischbacher, Urs (2007), “z-tree: Zurich toolbox for ready-made economic experiments.”Experimental Economics, 10, 171–178.

Keefe, R. (2007), Theories of Vagueness. Cambridge Studies in Philosophy, Cambridge Uni-versity Press.

Keefe, Rosanna and Peter Smith (1997), Vagueness: A Reader. MIT Press.

Lambie-Hanson, Timothy and Giri Parameswaran (2015), “A model of vague communica-tion.”

Lewis, David (1969), Convention: A Philosophical Study. Harvard University Press.

Lipman, Barton L. (2009), “Why is language vague.” Working paper.

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Pinker, Steven (2007), The stuff of thought: Language as a window into human nature.Penguin.

Quine, Willard V. O. (1960), Word and object. Technology Press of the Massachusetts In-stitute of Technology, Cambridge.

Serra-Garcia, Marta, Eric van Damme, and Jan Potters (2011), “Hiding an inconvenienttruth: Lies and vagueness.” Games and Economic Behavior, 73, 244 – 261.

Sobel, Joel (2015), “Broad terms and organizational codes.” Working paper.

van Deemter, Kees (2010), Vagueness Facilitates Search, 173–182. Springer Berlin Hei-delberg, Berlin, Heidelberg.

Wood, Daniel H. (2016), “Communication-enhancing vagueness.” Working paper.

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Appendices

A Experimental Instructions - Treatment 2A-U-IND

INSTRUCTION

Welcome to the experiment. This experiment studies decision making between threeindividuals. In the following two hours or less, you will participate in 21 rounds of decisionmaking. Please read the instructions below carefully; the cash payment you will receiveat the end of the experiment depends on how well you make your decisions according tothese instructions.

Your Role and Decision Group

There are 24 participants in today’s session. One third of the participants will be ran-domly assigned the role of Member A, another one third the role of Member B, and theremaining the role of Member C. Your role will remain fixed throughout the experiment.At the beginning of the first round, three participants, one Member A, one Member B andone Member C, will be matched to form a group of three. The three members in a groupmake decisions that will affect their rewards in all 21 rounds. That is, you will stay in thesame group so that you will interact with the same two other participants throughout the21 rounds. You will not be told the identity of the participants in your group, nor will theybe told your identity—even after the end of the experiment.

Your Decision and Earning in Each of Round 1-20

In each round and for each group, the computer will select two integer numbers A and Bbetween 0 and 100 randomly and independently. Each possible number has equal chanceto be selected. The selected number A will be revealed only to Member A and the selectednumber B will be revealed only to Member B. Member C, without seeing any of thesenumbers, will have to choose one of two actions UP and DOWN.

The amount of Experimental Currency Unit (ECU) you earn in a round depends on thetwo numbers A and B as well as the action chosen by Member C. In particular,

1. When A +B > 100, if Member C chooses

(a) UP, every member in your group will receive 50 ECU.

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(b) DOWN, every member in your group will receive 0 ECU.

2. When A +B < 100, if Member C chooses

(a) DOWN, every member in your group will receive 50 ECU.

(b) UP, every member in your group will receive 0 ECU.

3. When A+B = 100, every member in your group will receive 50 ECU regardless of theaction chosen by Member C.

Member A’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 5(a). The horizontal axis represents the number A and the vertical axis representsthe number B. You will see a blue vertical line, which represents the actually selectednumber A in the horizontal axis. The red diagonal line represents the cases with A +B =

100.

With all this information on your screen, you will be asked to send one of two messages“A is LOW” and “A is HIGH” to Member B in your group. Once you click one of the messagebuttons, your decision in the round is completed and your message will be transmitted toMember B in your group.

(a) Member A’s Screen (b) Member B’s Screen

Figure 5: Screen Shots

Member B’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 5(b). The horizontal axis represents the number A and the vertical axis representsthe number B. You will see a blue horizontal line, which represents the actually selectednumber B in the vertical axis. The red diagonal line represents the cases with A+B = 100.

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You will also receive a message from Member A in your group. With all this informationon your screen, you will be asked to send one of two messages “B is LOW” and “B is HIGH”to Member C in your group. Once you click one of the message buttons, your decision inthe round is completed and your message will be transmitted to Member C in your group.

Member C’s Decisions

You will be presented with a two-dimensional coordinate system on your screen as inFigure 6. The horizontal axis represents the number A and the vertical axis representsthe number B. The red diagonal line represents the cases with A +B = 100.

You will receive a message from Member B in your group. With all this information onyour screen, you will be asked to take one of two actions DOWN and UP. Once you clickone of the action buttons, your decision in the round is completed.

Figure 6: Member C’s Screen

Information Feedback

At the end of each round, the computer will provide a summary for the round: actuallyselected numbers A and B, Member A’s message, Member B’s message, Member C’s actionchoice, and your earning in ECU.

Your Decision in Round 21

After the 20th round, your screen will provide further instructions for your decisions inRound 21. The game you are going to play in this round is essentially the same as before,but you need to follow some new procedures. Please read the instructions carefully beforeyou start the 21st round. You will have an opportunity to ask questions if anything isunclear about the new instructions.

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Your Cash Payment

To calculate your cash payment, the experimenter will randomly select two rounds tocalculate your cash payment. Each round between Rounds 1 and 21 has an equal chanceto be selected. So it is in your best interest to take each round seriously. Your total cashpayment at the end of the experiment will be the sum of ECU you earned in the twoselected rounds, translated into HKD with the exchange rate of 1 ECU = 1 HKD, plus a30 HKD show-up fee.

Quiz and Practice

To ensure your understanding of the instructions, we will provide you with a quiz andpractice round. We will go through the quiz after you answer it on your own.

You will then participate in 1 practice round. The practice round is part of the in-structions which is not relevant to your cash payment; its objective is to get you familiarwith the computer interface and the flow of the decisions in each round. Once the practiceround is over, the computer will tell you “The official rounds begin now!”

Administration

Your decisions as well as your monetary payment will be kept confidential. Rememberthat you have to make your decisions entirely on your own; please do not discuss yourdecisions with any other participants. Upon finishing the experiment, you will receiveyour cash payment. You will be asked to sign your name to acknowledge your receipt ofthe payment. You are then free to leave. If you have any question, please raise your handnow. We will answer your question individually.

1. Suppose you are assigned to be a Member A. The computer chooses the random numbers A = 25 andB = 50. Which of the following is true?

(a) Both you and Member B know the chosen numbers A and B but Member C does not know anyof the numbers.

(b) Neither you nor Member B knows the chosen numbers A and B.

(c) You are the only person in your group who knows the chosen number A and Member B is theonly person in your group who knows the chosen number B.

2. Suppose that the computer chooses the random numbers A = 25 and B = 50. Member C in your grouptakes action DOWN. Please calculate the earning for each player:

• Member A’s payoff:

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• Member B’s payoff:

• Member C’s payoff:

3. Suppose that the computer chooses the random numbers A = 60 and B = 73. Member C in your grouptakes action DOWN. Please calculate the earning for each player:

• Member A’s payoff:

• Member B’s payoff:

• Member C’s payoff:

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B Scripts for Strategy-method and Belief-elicitation -Treatment 2A-U-IND

1. Strategy - Member A

In this round, we ask you to report your plan. After you specify your plan below, A will be realized and your plan will beimplemented accordingly.

Your plan: Send “A is LOW” if A is less than or equal to , and otherwise, send “A is HIGH”.What is the number for you in the blank above?

2. Belief - Member A

In this round, Member A is going to report his/her plan according to the following form:Send “A is LOW” if A is less than or equal to , and otherwise, send “A is HIGH”.

What do you think is the number for him/her in the blank above?If your guess is in the range of the actual value (chosen by Member A) plus-minus 5, then you will receive extra 8 ECU.

3. Strategy - Member B

In this round, we ask you to report your plan. After you specify your plan below, A and B will be realized and your plan willbe implemented accordingly.

(a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to , andotherwise, send “B is HIGH”.

(b) When receiving “A is HIGH” from Member A, send “B is LOW” if B is less than or equal to , andotherwise, send “B is HIGH”.

What is the number for you in the first blank above?What is the number for you in the second blank above?

4. Belief - Member B

In this round, Member B is going to report his/her plan according to the following form:(a) When receiving “A is LOW” from Member A, send “B is LOW” if B is less than or equal to , and

otherwise, send “B is HIGH”.(b) When receiving “A is HIGH” from Member A, send “B is LOW” if B is less than or equal to , and

otherwise, send “B is HIGH”.What do you think is the number for him/her in the blank in (a)?What do you think is the number for him/her in the blank in (b)?

If each of your guesses for (a) and (b) is in the range of the actual value (chosen by Member B) plus-minus 5, then you willreceive extra 4 ECU.

5. Strategy - Member C

In this round, we ask you to report your plan. After you specify your plan below, A and B will be realized and your plan willbe implemented accordingly.

What action would you like to take if the message from Member B is(a) B is LOW (b) B is HIGH

6. Belief - Member C

In this round, we ask Member C to report his/her plan about what action to take for each possible message.What action do you think would Member C like to take if the message from Member B is

(a) B is LOW (b) B is HIGHIf each of your guesses for (a) and (b) is correct, then you will receive extra 4 ECU.

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C Figures and Tables

(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 7: Alice’s Messages

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(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 8: Bob’s Messages ∣ “Low” from Alice

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(a) Treatment 2A-O-IND (b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Note: The red dotted lines indicate the optimal cut-off equilibrium strategy.

Figure 9: Bob’s Messages ∣ “High” from Alice

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Figure 10: Alice’s Elicited Strategy and Other Players’ Beliefs

Figure 11: Bob’s Elicited Strategy and Other Players’ Beliefs

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(a) Treatment 2A-O-IND

(b) Treatment 3A-O-IND

(c) Treatment 2A-U-IND (d) Treatment 3A-U-IND

(e) Treatment 2A-U-IMP (f) Treatment 3A-U-IMP

Figure 12: Charlie’s Elicited Strategy and Other Players’ Beliefs’

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